New transmission won't make any difference since that's not your issue.

There are several reasons why the truck is sluggish and most of it is the manufacturer design. They were trying to keep emissions and carb rating down and did the following to look good on paper but it doesn't perform in the real world.
The problem starts with a 6000lb truck and only 230 horsepower. (A decent amount for the time)
The manufacturer put 3.55 gears in keeping rpms way below the power curve.
The manufacturer engages TC lockup at 45mph, again taking motor out of power curve.
The manufacturer put in 1600 stall Torque Converter, again taking motor out of power curve.
The manufacturer programmed in a torque management to save transmission life.
Then you installed 33 inch tires on this truck that has been severely degraded by it's designers, again taking it way out of power curve.

Suggestions
Go back to stock tires and get 4.10 gears(what I have), this bumps rpm 400 over stock. Or get 4.56 gears and keep 33 inch tires, same equivalent. The truck will cruise at 2000 rpm at 65mph which is where a 6000lb truck with 230hp needs to be at.
Get a custom tune and move TC lockup to 55mph, and remove torque management.
Next time transmission is removed, install 2200 stall torque converter. This may produce more heat so install a trans temp gauge.

If you do this the truck will perform much better climbing hills and pulling while the engine is in the sweet spot, not revving too high but just high enough to begin to make power, even with only 230hp. There's no power to be made at 1600rpm or lower.

Good luck.

 

4.56 gears and 33" tires isn't a good match.... That's what I have, and I don't care for it. Towing on the freeway is unpleasant, at best, as the engine is pushing over 3K RPM at 65 mph. 4.10s and 33's I think would be just about perfect.
TCC engage is OK, but, O/D could kick in later......
96 doesn't have as much in the way of torque management as the 98 and up trucks, still, a tuner will most certainly help.

These engines do start developing a lot of their torque pretty low in the RPM range. And by the time you hit your 'high gear', (3rd or 4th, depending on if O/D is enabled or not....) you are going to be fairly close to your cruising speed anyway, and it doesn't take quite as much power to maintain your speed, as it does to accelerate to it.

The tuner will bump power a bit as well, not so much horsepower, as it will torque though. (dyno test in the faq section somewhere....)

Biggest thing for the OP though, I think would be gears. Of course, that's the most expensive upgrade as well..... (1500 - 2000 if you pay someone to do it.)

 

Thanks Heyyou, I meant OD engage to 55mph.

 

You can tow in OD as long as you can mantain speed without downshifting a lot. That will keep you from being at 3000 rpm when you don't need to be.

 

Dodge does not recommend towing in O/D AT ALL........ O/D is .69:1, a pretty significant drop, and even when unloaded, generates a significant amount of heat..... add a few thousand pounds to the equation, and the trans really isn't going to like that. Not to mention, I am usually pulling a 4000 pound trailer, with 6-8000 pounds of dirt/stuff in it.... O/D is simply not an option.

 

Cummins trucks come with the 47re and you almost have to tow in overdrive with those because of the diesels lower rpm power band. The 47re will hold up fine behind a stock Cummins towing in OD as long as you don't lug them too much, I don't see why it would be any different with the v10.